3. How large is the maximum field strength a
dielectric can take? Remember,
no material can take arbitrarily large loads - mechanical, electical,
whatever.

For starters, we look at some general
descriptions, definitions and general relations of the quantities necessary in
this context.

The dielectric constant of solids is
an interesting material parameter only if the material is exposed to an
electrical field (and this includes the electrical field of an
electromagnetic
wave). The effect of the electrical
field (or just field for short
from now on) can be twofold:

1. It induces
electrical dipoles in the material and tries to align them in the
field direction. In other words, with a
field, dipoles come into being that do not exist without a field.

2. It tries to align dipoles that are already present in the
material. In other words, the material contains
electric dipoles even without a field.

Of course we also may have a
combination of both effects: The electrical field may change the distribution
of existing dipoles while trying to align them, and it may generate new dipoles in addition.

The total effect of an electrical
field on a dielectric material is called the polarization of the material.

To understand that better, lets look
at the most simple object we have: A single atom (we do not even consider molecules at this
point).

We have a positively charged nucleus and the
electron "cloud". The smeared-out negative charge associated with the
electron cloud can be averaged in space and time, and its charge center of
gravity than will be at a point in space that coincides exactly with the location of the nucleus, because we
must have spherical symmetry for atoms.

If we now apply an electrical field, the centers
of charge will be separated. The electron cloud will be pulled in the direction
of the positive pole of the field, the nucleus to the negative one. We may
visualize that (ridiculously exaggerated) as follows:

The center of the positive and
negative charges q (= z · e) are now separated by a
distance x, and we thus induced a dipole momentm which is defined by

m = q ·
x

It is important to understand that m is a vector because x is a vector. The way we define it, its
tip will always point towards the positive
charge. For schematic drawings we simply draw a little arrow for
m.

The magnitude of this induced
dipole moment is a
property of our particular atom, or, if we generalize somewhat, of the
"particles" or building blocks of the material we are studying.

In order to describe the
bulk material - the sum of the particles - we
sum up all individual dipole moments
contained in the given volume of the material and divide this sum by the volume
V. This gives us
the (volume independent) polarizationP of the
material. Notice that we have a
vector
sum!

P thus points from the
negative to the positive charge, too - a convention opposite to that used for the electrical field.

The physical dimension of the polarization thus
is C/m2; (Coulomb
per square meter). i.e. the polarization has the dimension of an area
charge, and since m is a
vector, P is a vector, too.

It is important to realize that a polarization
P = 0 does not mean
that the material does not contain dipole
moments, but only that the
vector sum of all
dipole moments is zero.

This will always be the case if
the dipole moment vectors are randomly
distributed with respect to their directions. Look at the
picture in one of the next
subchapters if you have problems visualizing this. But it will also happen if
there is an ordered distribution with pairs of opposing dipole moments; again
contemplate a picture in one of the
following subchapters if that statment is not directly obvious.

That P has the
dimension of C/cm2, i.e. that of an area charge, is not accidental but has an immediate
interpretation.

To see this, let us consider a simple plate
capacitor or condenser with a homogeneously polarized
material inside its plates. More generally, this describes an
isotropic dielectric slab of material in a homogeneous electrical field. We
have the following idealized situation:

For sake of simplicity, all dipole moments have
the same direction, but the subsequent reasoning will not change if there is
only an average component of P in field direction. If we
want to know the charge densityr inside a small probing volume, it is
clearly zero in the volume of the material (if averaged over probing volumes
slightly larger than the atomic size), because there are just as many positive
as negative charges.

We are thus left with the surfaces, where there is indeed some charge as
indicated in the schematic drawing. At one surface, the charges have
effectively moved out a distance x , at the
other surface they moved in by the same amount. We thus have a
surface charge, called a
surface polarization charge, that we can
calculate.

The number
Nc of charges appearing at a surface with area
A is equal to the number of dipoles contained in the surface
"volume" VS = A · x times the relevant charge q of the dipole.
Using x to define the volume makes sure that
we only have one layer of dipoles in the
volume considered.

Since we assume a homogeneous dipole
distribution, we have the same polarization in any volume and thus
P = Sm / V = SmS / VS obtains.
Therfore we can write

P =

SVmV

=

SSmVS

=

x · SSqVS

=

x · SSqx · A

=

SSqA

SSq =

Nc =

P · A

SV
or SS denotes that the summation
covers the total volume or the "surface" volume. Somewhere we
"lost" the vector property of P, but that only
happens because we automatically recognize x
as being perpendicular to the surface in question.

While this is a certain
degree of
sloppiness, it makes life much
easier and we will start to drop the underlining of the vectors from now on
whenever it is sufficiently clear what is meant.

Of course, spol is a scalar, which we obtain if we consider
P · A to be the scalar product of the vector
P and the vector A; the
latter being perpendicular to
the surface A with magnitude |A| =
A.

In purely
electrical terms we thus can always replace a material with a homogeneous polarization P by
two surfaces perpendicular to some direction, - lets say z - with
a surface charge density of Pz (with, of course,
different signs on the two different surfaces).

If the polarization vector is
not perpendicular to the surface we chose,
we must take the component of the
polarization vector parallel to the normal
vector of the surface considered . This is automatically taken care
of if we use the vector formulation for A.

A dielectric material now quite
generally reacts to the presence of an electrical field by becoming polarized
and this is expressed by the direction and magnitude of P.

P is a measurable quantity tied to the specific material
under investigation. We now need a material
law that connects cause and effect, i.e. a relation between the the
electrical field causing the polarization and the amount of polarization
produced.

Finding this law is of course the task of basic
theory. But long before the proper theory was found, experiments supplied a
simple and rather (but not quite) empirical
"law":

If we measure the
polarization of a material, we usually find a linear relationship between the
applied field E and P, i.e.

Note that including e0 in the relation is a convention which is
useful in the SI system, where
charges are always coupled to electrical fields via e0. There are other systems, however,
(usually variants of the
cgs
system), which are still used by many and provide an unending source of
confusion and error.

This equation is to dielectric
material what Ohms law
is to conductors. It is no more a real "law of
nature" than Ohms law, but a description of many experimental
observations for which we will find deeper reasons forthwith.

Our task thus is to calculatec from
basic material parameters.

Connection between the Polarization
P and the Electrical Displacement D

Next we need the
connection between the polarization P, or the dielectric
susceptibility c, with some older quantities
often used in connection with
Maxwellsequations.

Note that in the English literature
often the abbreviation k ("Kappa")
is used; in proper microelectronics slang one than talks of "low k
materials" (pronounced "low khe" as in (O)K) when one actually
means "low kappa" or "low epsilon relative".

D is supposed to give
the "acting" flux inside the
material.

While this was a smart thing to do
for Maxwell and his contemporaries, who couldn't know anything about materials
(atoms had not been "invented" then); it is a bit unfortunate in
retrospect because the basic quantity is
the polarization, based on the elementary
dipoles in the material, and the material
parameterc describing this - and
not some changed "electrical flux density" and the relative
dielectric constant of the material.

It is, however, easy (if slightly
confusing) to make the necessary connections. This is most easily done by
looking at a simple plate capacitor. A full treatise is found in a
basic module, here we just
give the results.

Theelectric
displacementD in a dielectric caused by
some external field Eex is the displacement
D0 in vacuum plus the
polarizationP of the material, i.e.

D = D0 + P
= e0 · E + P

Inserting everything
we see that the relative dielectric constant er is simply the dielectric susceptibilityc plus 1.

er =

1 + c

For this
"translations" we have used the relation P =
e0 · c
· E, which isnot an
a priori law of nature, but an
empirical relation. However, we are going to prove this relation for specific,
but very general cases forthwith and thus justify the equations above.

We have also simply implied that
P is parallel to E, which is only reasonable for isotropic materials.

Inanisotropic media, e.g. non-cubic crystals,
P does not have to be parallel to E, the scalar quantities er and c
then are tensors.

The basic task in the materials
science of dielectrics is now to calculate (the tensor) c from "first principles", i.e. from basic
structural knowledge of the material considered. This we will do in the
following paragraphs.